Systems and methods for noise tolerant signal processing in pilot assisted data receivers

ABSTRACT

The present disclosure provides systems and methods for noise tolerant signal processing in a pilot assisted data receiver, including: given received pilots with common pilot components and individual pilot components, computing coefficients associated with the individual pilot components of the received pilots; and applying the computed coefficients to the received pilots to obtain conditioned pilots. The individual pilot components result from relatively slow changes of the received pilots relative to the common pilot components. The common pilot components result from relatively fast changes of the received pilots relative to the individual pilot components.

FIELD OF THE DISCLOSURE

The present disclosure relates generally to systems and methods for usein communication networks, such as fiber optic communication networks,digital subscriber line (DSL) communication networks, wirelesscommunication networks, and the like. More specifically, the presentdisclosure relates to systems and methods for noise tolerant signalprocessing in pilot assisted data receivers.

BACKGROUND OF THE DISCLOSURE

Fiber optic communication networks and the like are experiencing rapidlyincreasing capacity growth. This capacity growth is reflected inindividual channel data rates inexorably scaling from 10 Gbps, to 40Gbps, to 100 Gbps, to 1000 Gbps channels, and so on. The capacity growthis also reflected in increasing total channel counts carried within anoptical fiber, for example.

The ever growing demand for increased bandwidth and channel capacity isbeing met by broadband polarization division multiplexed (PDM) coherentsystems and the like utilizing multi lever amplitude/phase modulationformats, for example. In such systems, the essential tasks ofpre-conditioning the signal at the transmitter and signal processingafter coherent detection are carried out by extremely fast digitalsignal processing (DSP) integrated circuit (IC) chips. The tendencytowards dramatic growth of the DSP chips'size, complexity, price, andpower consumption and dissipation with ever increasing processing speedsis well researched and thoroughly documented.

A significant portion of the DSP algorithms and computing power isdevoted to the solution of two crucial and computationally intensetasks: polarization de-multiplexing of received signals and laserfrequency offset recovery and phase noise cancellation. The formerinvolves the estimation, calculation, and application of the elements ofthe inverse Jones matrix of the optical fiber link for the recovery ofthe originally linearly polarized signals, while the later involves thedetermination of the fast spinning phasor generated by the beating ofthe signal carrier and local oscillator (LO) lasers and its utilizationfor phase locking of the coherently detected signals and rectifying themfor the following steps of the data recovery processes. Thesecomputationally intense algorithms are accountable for the majority ofthe DSP chips' real estate and power usage.

As an alternative, pilot assisted techniques designed to perform theabove tasks were conceived at the early onset of coherent fiber opticcommunication system design, long before the DSP era. The un-modulated,but either frequency shifted or orthogonally polarized, portion of thesignal carrier laser light was launched into the optical fiber alongwith the information carrying signal and served as a built in phasenoise reference at the receiver. Down mixing of the signal with thepilot at the receiver (i.e. multiplying the signal W by the complexconjugate pilot P*: W×P*) canceled their common noisy phase factorsoriginating from the random phase walk due to finite linewidth of thesignal carrier and LO lasers, as well as de-multiplexing thepolarizations.

These pilot based signal processing techniques re-surfaced recently,allowing not only for much simpler ways of cancelling laser frequencyoffset and phase noise, but also for compensating the nonlinear phasedistortion, greatly reducing or even eliminating the need for fast,expensive, and power hungry digital electronic signal processing.

However, all of the existing pilot assisted techniques suffer from onecommon drawback: the noise accompanying the pilot, mostly amplifiedspontaneous emission (ASE) noise generated in optical amplifiers,significantly compromises the pilot's ability to perform its functions,resulting in a pilot related signal quality (Q) penalty. Specialmeasures are required to reduce or even eliminate the pilot noiserelated Q penalty.

BRIEF SUMMARY OF THE DISCLOSURE

The present disclosure provides special averaging (in the time domain)or filtering (in the frequency domain) methods to ensure virtuallynoiseless pilot operation, while performing polarization de-multiplexingand laser phase noise cancellation in polarization multiplexed coherentsystems, as well as other systems, optionally employing wide linewidthdistributed feedback (DFB) signal and LO lasers (in the case of opticallinks). The averaging/filtering methods of the present disclosure areapplicable for analog signal processing (ASP) and DSP. It will bereadily apparent to those of ordinary skill in the art that theaveraging/filtering methods of the present disclosure are alsoapplicable to electrical and wireless links.

Long time averaging and narrow frequency filtering are conventionalmethods for reducing signal corrupting noise. But these methods are notreadily applicable to the pilots in pilot assisted coherent fiber opticcommunication systems, for example, due to the contraindications presentin the requirements for pilot averaging and filtering. The tightfiltering needed for noise reduction is in contradiction with the muchbroader spectral content of the pilots related to laser frequency offsetand phase noise and, more importantly, the nonlinear phase noise, whichrequires hundreds of MHz of pilot bandwidth to be compensated.

The central concept of the present disclosure is based on the differencein the physical mechanisms affecting the pilots. One of them isdetermined by the co-propagation of the pilots together with the signalalong the communication link all the way from the signal carrier laserto the intradyne coherent detector. Co-propagating pilots undergo thesame polarization evolution as does the signal and hence attain all theinformation about the Jones matrix of the fiber link, resulting indifferent pilot components being proportional to the correspondingmatrix elements (there are two per transmitted pilot, i.e. one for eachof the Jones matrix elements). This mechanism is responsible for changesin the individual pilot components, which occur slowly, especiallycompared to the other much more dynamic phenomenon, such that they maybe considered as quasi-static. The much faster dynamic mechanisminvolves the laser frequency offset and phase noise. These result in afast spinning phasor factor generated by the beating of the signalcarrier and LO lasers with the addition of a random phase walk from thelaser phase noise. This phasor factor is common to all pilot components,and together with different individual components, they describe thepilots completely.

This phenomenological subdivision of the pilot into individual staticand common dynamic components provides the avenue for accomplishing thedesired averaging/filtering. This is done by averaging/filteringindividual static pilot components separately from their common dynamiccounterparts. This accomplishes the majority of the noise cleaning task.Additionally, the common dynamic part of the pilots has the noiserelated fluctuations of its amplitude averaged/filtered out, convertingit to a constant amplitude phasor. The combination of these measuresresults in improved averaged/filtered pilots which accomplish the tasksof phase compensation and polarization de-multiplexing in spite of thepresence of ASE generated noise underneath.

Thus, the systems and methods of the present disclosure resolve thecontraindications in pilot based signal processing: the need fornarrowband filtering to mitigate ASE noise penalties, and the need forwide spectral bandwidth to mitigate laser frequency, laser linewidth,and nonlinear impairments.

The systems and methods of the present disclosure provide the followingbenefits, among others:

-   -   1. the disclosed pilot averaging/filtering technique provides        near complete elimination of performance deteriorating noise        from pilots;    -   2. the disclosed pilot averaging/filtering technique restores        the data channel Q factor values back to the level corresponding        to noiseless pilots;    -   3. the disclosed pilot averaging/filtering technique preserves        the benefits of pilot based processing in eliminating wideband        laser phase noise;    -   4. the disclosed pilot averaging/filtering technique preserves        the benefit of pilot based processing by substantially        simplifying coherent receiver algorithms, and reducing        associated electronic circuit size, cost, and power; and    -   5. the disclosed pilot averaging/filtering technique preserves        the benefit of wideband pilot based processing in mitigating        nonlinear optical fiber propagation impairments.

The uniqueness of the disclosed techniques is in their non-trivialimplementation of pilot averaging/filtering, allowing for rigorousreduction of noise via averaging/filtering, while maintaining theirbroadband phase noise cancelling capabilities. This distinctiveaveraging/filtering of static pilot components and averaging/filteringof the dynamic phasors' magnitude are accomplished without theoverburdening labor of separating them before and re-combining themafter the averaging/filtering procedures.

In one exemplary embodiment, the present disclosure provides a methodfor noise tolerant signal processing in a pilot assisted data receiver,including: given received pilots with common pilot components andindividual pilot components, computing coefficients associated with theindividual pilot components of the received pilots; and applying thecomputed coefficients to the received pilots to obtain conditionedpilots. The individual pilot components result from relatively slowchanges of the received pilots relative to the common pilot components.The common pilot components result from relatively fast changes of thereceived pilots relative to the individual pilot components. The methodalso includes removing noise from the individual pilot components byaveraging/filtering the individual pilot components. Theaveraging/filtering the individual pilot components includes one ofaveraging the individual pilot components in a time domain and low passfiltering the individual components in a frequency domain. The methodfurther includes preserving the common pilot components using widebandsignal processing. The method still further includes down converting thereceived pilots to a baseband. Applying the computed coefficients to thereceived pilots to obtain the conditioned pilots includes deriving alinear combination of the pilot components with weights corresponding tothe computed coefficients. The method still further includes convertingthe conditioned pilots to a constant amplitude phasor by removingassociated noise related to amplitude fluctuations.

In another exemplary embodiment, the present disclosure provides a pilotassisted data receiver for noise tolerant signal processing, including:a processing block operable for, given received pilots with common pilotcomponents and individual pilot components, computing coefficientsassociated with the individual pilot components of the received pilots;and a processing block operable for applying the computed coefficientsto the received pilots to obtain conditioned pilots. The individualpilot components result from relatively slow changes of the receivedpilots relative to the common pilot components. The common pilotcomponents result from relatively fast changes of the received pilotsrelative to the individual pilot components. The data receiver alsoincludes a processing block operable for removing noise from theindividual pilot components by averaging/filtering the individual pilotcomponents. The averaging/filtering the individual pilot componentsincludes one of averaging the individual pilot components in a timedomain and low pass filtering the individual components in a frequencydomain. The data receiver further includes a processing block operablefor preserving the common pilot components using wideband signalprocessing. The data receiver still further includes a processing blockoperable for down converting the received pilots to a baseband. Applyingthe computed coefficients to the received pilots to obtain theconditioned pilots includes deriving a linear combination of the pilotcomponents with weights corresponding to the computed coefficients. Thedata receiver still further includes a processing block operable forconverting the conditioned pilots to a constant amplitude phasor byremoving associated noise related to amplitude fluctuations.

In a further exemplary embodiment, the present disclosure provides amethod for noise tolerant signal processing in a pilot assisted datareceiver, including: down converting received pilots to a baseband;given received pilots with common pilot components and individual pilotcomponents, computing coefficients associated with the individual pilotcomponents of the received pilots; removing noise from the individualpilot components by averaging/filtering the individual pilot components;preserving the common pilot components using wideband signal processing;applying the computed coefficients to the received pilots to obtainconditioned pilots; and converting the conditioned pilots to a constantamplitude phasor by removing associated noise related to amplitudefluctuations. The averaging/filtering the individual pilot componentsincludes one of averaging the individual pilot components in a timedomain and low pass filtering the individual components in a frequencydomain, and applying the computed coefficients to the received pilots toobtain the conditioned pilots includes deriving a linear combination ofthe pilot components with weights corresponding to the computedcoefficients.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is illustrated and described with reference tothe various drawings, in which like reference numbers are used to denotelike system components/method steps, as appropriate, and in which:

FIG. 1 is an optical spectra plot of a single carrier PDM QPSK signal,including horizontal and vertical polarizations with inserted horizontaland vertical pilot carriers at the nearest notch(es);

FIG. 2 is an optical spectra plot of a multiple carrier (e.g. ninesubcarrier) OFDM QPSK signal, including horizontal and verticalpolarizations with inserted horizontal and vertical pilot carriers justoutside of the signal spectrum;

FIG. 3 is a schematic diagram illustrating one exemplary embodiment ofthe polarization and phase diverse intradyne coherent detection and RFbaseband down conversion of pilots and signals in accordance with thesystems and methods of the present invention;

FIG. 4 is a series of plots illustrating noisy pilots' traces in thecomplex plane: (a)—P_(X) ^(H), (b)—P_(X) ^(V), (c)—P_(Y) ^(H), and(d)—P_(Y) ^(V), raw pilots and stabilized phasor amplitudes inaccordance with Equation (3) below;

FIG. 5 is a schematic diagram illustrating one exemplary embodiment ofthe computations performed to generate conditioned and conjugated pilotsin accordance with the systems and methods of the present invention;

FIG. 6 is a series of plots illustrating averaged/filtered and noisypilot traces in the complex plane: (a)—P_(X) ^(H), (b)—P_(X) ^(V),(c)—P_(Y) ^(H), and (d)—P_(Y) ^(V), raw pilots, static parts of thepilots averaged/filtered by pre-conditioning, additionally stabilizedphasor amplitudes in accordance with Equation (3) below, +—a prioriknown ideal static parts of the pilots as used in simulations (top lineof the titles), and ×—extracted from raw pilot data static parts of thepilots (bottom line of the titles); and

FIG. 7 is a plot illustrating the Q factor (in dB) when computed usingthe raw pilots, the pre-conditioned pilots, the rectified phasors, andnoiseless pilots.

DETAILED DESCRIPTION OF THE DISCLOSURE

Conventional pilot assisted signal processing techniques were conceivedmore than two decades ago, during the early days of the development ofcoherent optical communication systems, long before the advent of erbiumdoped fiber amplifiers (EDFAs) and fast DSP. Optical carrier pilots weregenerated by either frequency shifting or flipping the polarization of asignificant portion (usually half) of the total transmitted laser power,which could be either opposite phase modulated or frequency shifted.

By co-propagating along with the information loaded signal all the waythrough the fiber optic link, the carrier pilot serves as a built inphase reference to and polarization replica of the signal at thedetection point. After self-coherent or coherent detection, pilot andsignal are down mixed electronically by applying them to the mixer andlow pass filtering (LPF). This procedure is mathematically equivalent tothe multiplication of the signal W by complex conjugate pilot P*: W×P*,which completely cancels the common phase factors comprised of thefrequency beatings and the phase noise of the signal and LO lasers. Thiscancellation greatly relaxes laser linewidth requirements, thus allowingthe use of a multi-MHz broad linewidth DBF laser as a signal carrier andLO in a coherent fiber optic communication system, for example. The useof the inherent polarization transformation sensing properties of thecarrier pilots allows the detection of the Stokes parameters of thefiber link and the unwrapping of the polarization of the signal byinverting the Jones matrix of the optical fiber.

Recently, the interest in pilot assistance was revived when pilot basedself-homodyne phase shift keying (PSK), quadrature phase shift keying(QPSK), and coherent wavelength division multiplexing (WDM) polarizationmultiplexed systems were introduced. These use roughly half of theoptical carrier's power as a polarization multiplexed pilot carrier. Atthe receiver, the pilot carrier is polarization de-multiplexed from thesignal and realigned with it at the photodetector, serving as aco-propagating LO for homodyne coherent detection with all of the abovementioned benefits related to inherent laser phase noise immunity.Self-heterodyne detection has been used with optical frequency divisionmultiplexing (OFDM) coherent systems, where a radio frequency (RF)synthesized pilot was located outside of the signal spectrum, instead ofbeing polarization multiplexed, and was separated from the signal bytight optical filtering before mixing at the photodetector. Bothself-homodyne and self-heterodyne systems eliminate the need for phaseand polarization diversity of the receiver by analog down mixing of theco-polarized signal and the pilot carrier having intrinsically identicaland mutually cancelling phase noise factors. This comes with the priceof the necessity to share a significant portion (roughly half) of thetotal optical power between the signal and pilot, resulting insignificant Q penalty, in spite of elaborate filtering, amplification,and optimization.

Modern coherent optical OFDM systems make further use of pilotassistance by utilizing much higher stability RF oscillators andfrequency selectivity of the electrical filters as compared to theiroptical counterparts. Analog RF and microwave techniques aresuccessfully implemented both for pilot synthesis and insertion at thetransmitter and for filtering and down mixing with the signal at thereceiver in the RF domain. This allows the allocation of a much smallerportion of the optical power for the pilot carrier, preserving theoptical power budget of the signal. Compared to self-coherent systems,the use of a designated powerful LO laser for coherent detection at thereceiver boosts coherent detection gain significantly.

Advanced electronic techniques not only refine the laser noisecancelling benefits of pilot assistance, but also contribute to thediscovery of non-linear phase noise cancellation capabilities inself-homodyne and RF-pilot based coherent optical OFDM systems. Pilotassisted techniques have been reported to cancel self phase modulation(SPM) and compensate cross phase modulation (XPM) originated non-linearphase noise. The broadband spectral content of the non-linear phasenoise component to be compensated requires significant, up to hundredsof MHz, pilot bandwidth for efficient non-linear phase noisecompensation, unfortunately capturing significant ASE noise under thepilot, leading to sacrificed performance.

All of the above referenced benefits of pilot assisted fiber optictransmission, for example, have specific pilot related drawbacks.Self-homodyne and self-heterodyne systems are forced to split theoptical power budget between the signal and pilot, sacrificing thesignal power and optical signal-to-noise ratio (OSNR) performance.RF-pilot based coherent OFDM systems may have reduced pilot powerrequirements, but their performance also suffers from the detrimentaleffects of the ASE noise accumulated under the pilot spectrum,especially when broadband non-linear phase noise compensation techniquesare utilized. The development and implementation of specialaveraging/filtering techniques allows for the reduction and evenelimination of the detrimental effects of pilot noise.

Thus, modern intradyne coherent PDM fiber optic communication systemsmay benefit greatly from carrier pilot assistance in performing suchtasks as laser frequency offset compensation, laser phase noisecancellation, non-linear phase noise compensation, and polarizationde-multiplexing. Pilots make this processing much simpler, faster, andmore efficient.

Referring specifically to FIG. 1, the optical spectra of a singlecarrier PDM QPSK signal 10 is illustrated, including horizontal 12 andvertical 14 polarizations with inserted horizontal 16 and vertical 18pilot carriers at the nearest notch(es). Referring specifically to FIG.2, the optical spectra of a multiple carrier (e.g. nine subcarrier) OFDMQPSK signal 20 is illustrated, including horizontal 12 and 22 andvertical 14 and 24 polarizations with inserted horizontal 26 andvertical 28 pilot carriers just outside of the signal spectrum. Itshould be noted that the signals 10 and 20 and subcarriers 12, 14, 22,and 24 are relatively broad because of the data being carried, while thepilots 16, 18, 26, and 28 are narrowband spikes on opposite sides of thespectra 10 and 20, such that they are separable.

Referring specifically to FIG. 3, while omitting the technical detailsof the generation and insertion of such pilots 16, 18, 26, and 28, viaRF up conversion, optical signal side band modulation, and digitalsignal processing, for example, well known to those of ordinary skill inthe art, the horizontally (H) 12 and 22 and vertically (V) 14 and 24polarized PDM optical signals 10 and 20 (signal 30) are detected afterpropagation over a fiber optic link by polarization and phase diverseintradyne coherent photodetection 34 utilizing an LO 32 and hybrid 36,producing two complex electrical signals X 38 and Y 40, corresponding totwo perpendicular polarizations of the LO laser. Standard RF mixing 42and LPF filtering 44 techniques down convert the pilots 16, 18, 26, and28 and information carrying signals 22 and 24 to the baseband forfurther processing. Baseband signals 12 and 14 are not down converted atthis stage. All baseband pilots (P) and information carrying signals (W)are complex in phase and quadrature values, as a result of the RF downconversions. Subscripts indicate their appearance at two mutuallyorthogonal (X and Y) parts of the polarization diverse optical receiver,while superscripts indicate the pilots' origins at the horizontal (H) orvertical (V) polarizations of the transmitted optical signal. It isnoted that two original pilots produce four outputs due to misalignmentbetween the HV and XY principal polarization states.

The four pilots are comprised of their corresponding elements of theJones matrix (J₁₁, J₁₂, J₂₁, and J₂₂) describing the fiber optic linkand common phasor factor produced by laser frequency offset Δω to andphase noise φ(t), with the omission of all irrelevant constants:

P _(X) ^(H) ˜J ₁₁ e ^(i(Δωt+φ(t))) ;P _(X) ^(V) ˜J ₁₂ e ^(i(Δωt+φ(t)));P _(Y) ^(H) ˜J ₂₁ e ^(i(Δωt+φ(t))) ;P _(Y) ^(V) ˜J ₂₂ e^(i(Δωt+φ(t)))  (1)

Signal polarization de-multiplexing, laser frequency offsetcompensation, and laser phase noise cancellation are performed at thepost-detection processing stage by combining the information carryingsignals W_(X) and W_(Y) multiplied by the complex conjugate pilots,which now have inverse Jones matrix elements and phasors inverted withrespect to those of the signals W_(X) and W_(Y):

W _(X) ×P _(X) ^(H) *+W _(Y) ×P _(Y) ^(H) *=I ^(H) +jQ ^(H)

W _(X) ×P _(X) ^(V) *+W _(Y) ×P _(Y) ^(V) *=I ^(V) +jQ ^(V)  (2)

As a result, the right-hand sides of the above two equations representthe recovered in-phase and quadrature tributaries of the transmittedQPSK data carrying horizontal (H) and vertical (V) signals ready forsampling and decision thresholding.

As already stated, the noise under the pilots has a detrimental effecton their ability to perform the post-detection processing. This isillustrated in FIG. 4, where raw noisy pilots 100 are plotted in thecomplex plane. The perfectly circular phasors of noiseless pilots areturned into shaggy rings by noise, with the annulus collapsing forsmaller pilot amplitudes caused by diminishing Jones matrix elements, asin the cases of P_(X) ^(H) in (a) and P_(Y) ^(V) in (d) 102.

Balancing the need to eliminate the noise from under the pilots and topreserve the broadband phasor, a straightforward attempt ataveraging/filtering may be performed by stabilizing the amplitude of thepilots, while leaving the phase intact. This may be done by replacingthe pilots with their amplitude-stabilized versions:

$\begin{matrix}{{< P_{X,Y}^{H,V}>= < {P_{X,Y}^{H,V}} > {\times \frac{P_{X,Y}^{H,V}}{P_{X,Y}^{H,V}}}},} & (3)\end{matrix}$

where <o> designate averaging and |o| designate absolute value. Theabove formula expresses mathematically the transformation of shaggyannular pilots into constant radius circles with intact phase. Thisworks well for strong pilots, such as in the cases of PIT in (b) andP_(Y) ^(H) in (c) 104, but for weaker pilots with collapsed annularrings 102, this causes undue fast phase jumps when pilots cross near theorigin, as it is illustrated in the crisscrossing traces of (a) and (d)106, thereby preventing successful signal recovery.

To avoid the problem of collapsing annular pilots, the above describedamplitude stabilization procedure is preceded by a pre-conditioningaveraging/filtering stage, thereby eliminating the possibility ofannular pilot traces collapsing.

This pre-conditioning averaging/filtering stage is based on the factthat the pilots are comprised of a common fast spinning dynamic phasefactor produced by the laser frequency offset Δω to and phase noise φ(t)and the individual “quasi-static” slowly changing (in time) partsrelated to the polarization evolution and described by the Jones matrixelements. These are designated as “static” (S) and “dynamic” (D):

P _(X) ^(H) =D×S _(X) ^(H) ;P _(X) ^(V) =D×S _(X) ^(V) ;P _(Y) ^(H) =D×S_(Y) ^(H) ;P _(Y) ^(V) =D×S _(Y) ^(V).  (4)

By multiplying the pilots by their own complex conjugate static partsand realizing that the root mean square of the pilot absolute valuesgives the absolute value of their static parts:

P _(X) ^(H) ×S _(X) ^(H) *=D×|S _(X) ^(H)|² =D×<|P _(X) ^(H)|²>

P _(X) ^(V) ×S _(X) ^(V) *=D×|S _(X) ^(V)|² =D×<|P _(X) ^(V)|²>

P _(Y) ^(H) ×S _(Y) ^(H) *=D×|S _(Y) ^(H)|² =D×<|P _(Y) ^(H)|²>

P _(Y) ^(V) ×S _(Y) ^(V) *=D×|S _(Y) ^(V)|² =D×<|P _(Y) ^(V)|²>  (5)

From this, the expression for the common dynamic part of the pilots isderived:

$\begin{matrix}{D = \frac{{P_{X}^{H} \times S_{X}^{H^{*}}} + {P_{Y}^{H} \times S_{Y}^{H^{*}}} + {P_{X}^{V} \times S_{X}^{V^{*}}} + {P_{Y}^{V} \times S_{Y}^{V*}}}{< {{P_{X}^{H}}^{2} + {P_{Y}^{H}}^{2} + {P_{X}^{V}}^{2} + {P_{Y}^{V}}^{2}} >}} & (6)\end{matrix}$

The pre-conditioned complex conjugate pilots used in Equation (2) forpolarization de-multiplexing, laser frequency offset compensation, andlaser phase noise cancellation may be expressed by substituting Equation(6) into Equation (4):

<P _(X) ^(H) *>=|S _(X) ^(H)|² P _(X) ^(H) *+S _(X) ^(H) *S _(X) ^(V) P_(X) ^(H) *+S _(X) ^(H) *S _(Y) ^(H) P _(Y) ^(H) *+S _(X) ^(H) *S _(Y)^(V) P _(Y) ^(V)*

<P _(X) ^(v) *>=S _(X) ^(v) *S _(X) ^(H) P _(X) ^(H) *+|S _(X) ^(V)|² P_(X) ^(V) *+S _(X) ^(V) *S _(Y) ^(H) P _(Y) ^(H) *+S _(X) ^(V) *S _(Y)^(V) P _(Y) ^(V)*

<P _(Y) ^(H) *>=S _(Y) ^(H) *S _(X) ^(H) P _(X) ^(H) *+S _(Y) ^(H) *S_(X) ^(V) P _(X) ^(H) *+|S _(Y) ^(H)|² P _(Y) ^(H) *+S _(Y) ^(H) *S _(Y)^(V) P _(Y) ^(V)*

<P _(Y) ^(V) *>=S _(Y) ^(V) *S _(X) ^(H) P _(X) ^(H) *+S _(Y) ^(V) *S_(X) ^(V) P _(X) ^(V) *+S _(Y) ^(V) *S _(Y) ^(H) P _(Y) ^(H) *+|S _(Y)^(V)|² P _(Y) ^(V)*  (7)

The constant common denominator of the expression of Equation (6) for Dis omitted for simplicity, keeping in mind that it may always beabsorbed by the static parts of the pilots being normalized by it. Thisdenominator represents the total optical power of the pilots, and isconstant if the input signal power in the receiver is kept constant.

A 4-by-4 matrix S of constant coefficients involved in the calculationof the pre-conditioned pilots from the raw ones consists of the crossproducts of the still unknown static parts of the pilots and theircomplex conjugates:

$\begin{matrix}{S = \begin{pmatrix}{S_{X}^{H}}^{2} & {S_{X}^{H^{*}}S_{X}^{V}} & {S_{X}^{H^{*}}S_{Y}^{H}} & {S_{X}^{H^{*}}S_{Y}^{V}} \\{S_{X}^{V^{*}}S_{X}^{H}} & {S_{X}^{V}}^{2} & {S_{X}^{V^{*}}S_{Y}^{H}} & {S_{X}^{V^{*}}S_{Y}^{V}} \\{S_{Y}^{H^{*}}S_{X}^{H}} & {S_{Y}^{H^{*}}S_{X}^{V}} & {S_{Y}^{H}}^{2} & {S_{Y}^{H^{*}}S_{Y}^{V}} \\{S_{Y}^{V^{*}}S_{X}^{H}} & {S_{Y}^{V^{*}}S_{X}^{V}} & {S_{Y}^{V^{*}}S_{Y}^{H}} & {S_{Y}^{V}}^{2}\end{pmatrix}} & (8)\end{matrix}$

The process of computing both the S matrix of Equation (9) below and thepre-conditioned pilots of Equation (10) below is illustrated in FIG. 5and explained as follows. Similar to the equality of the root meansquare of the pilot absolute value to the absolute value of its staticpart used in Equation (5), the cross products of the still unknownstatic parts of the pilots and their complex conjugates in the matrix Smay be replaced by the averaged cross products of the pilots and theircomplex conjugates. This is due to the fact that D is a common phasefactor for all components, and drops out after conjugate multiplication.These are readily computable, thus eliminating the need to find thestatic parts of the pilots themselves:

$\begin{matrix}{S = \begin{pmatrix}{< {P_{X}^{H}}^{2} >} & {< {P_{X}^{H^{*}}P_{X}^{V}} >} & {< {P_{X}^{H^{*}}P_{Y}^{H}} >} & {< {P_{X}^{H^{*}}P_{Y}^{V}} >} \\{< {P_{X}^{V^{*}}P_{X}^{H}} >} & {< {P_{X}^{V}}^{2} >} & {< {P_{X}^{V^{*}}P_{Y}^{H}} >} & {< {P_{X}^{V^{*}}P_{Y}^{V}} >} \\{< {P_{Y}^{H^{*}}P_{X}^{H}} >} & {< {P_{Y}^{H^{*}}P_{X}^{V}} >} & {< {P_{Y}^{H}}^{2} >} & {< {P_{Y}^{H^{*}}P_{Y}^{V}} >} \\{< {P_{Y}^{V^{*}}P_{X}^{H}} >} & {< {P_{Y}^{V^{*}}P_{X}^{V}} >} & {< {P_{Y}^{V^{*}}P_{Y}^{H}} >} & {< {P_{Y}^{V}}^{2} >}\end{pmatrix}} & (9)\end{matrix}$

Using this version of the S matrix, the pre-conditioned pilots may beexpressed in terms of averaged cross products of the pilots and theircomplex conjugates (instead of their unknown static parts):

<P _(X) ^(H) *>=<|P _(X) ^(H)|² >P _(X) ^(H) *+<P _(X) ^(H) *P _(X)^(V) >P _(X) ^(V) *+<P _(X) ^(H) *P _(Y) ^(H) >P _(Y) ^(H) *+<P _(X)^(H) *P _(Y) ^(V) >P _(Y) ^(V)*

<P _(X) ^(V) *>=<P _(X) ^(V) *P _(X) ^(H) >P _(X) ^(H) *+|<P _(X)^(V)|² >P _(X) ^(V) *+<P _(X) ^(V) *P _(Y) ^(H) >P _(Y) ^(H) *+<P _(X)^(V) *S _(Y) ^(V) >P _(Y) ^(V)*

<P _(Y) ^(H) *>=<P _(Y) ^(H) *P _(X) ^(H) >P _(X) ^(H) *+<P _(Y) ^(H) *P_(X) ^(V) >P _(X) ^(V) *+<|P _(Y) ^(H)|² >P _(Y) ^(H) *+<P _(Y) ^(H) *P_(Y) ^(V) >P _(Y) ^(V)*

<P _(Y) ^(V) *>=<P _(Y) ^(V) *P _(X) ^(H) >P _(X) ^(H) *+<P _(Y) ^(V) *P_(X) ^(V) >P _(X) ^(V) *+<P _(Y) ^(V) *P _(Y) ^(H) >P _(Y) ^(H) *+<|P_(Y) ^(V)|² >P _(Y) ^(V)*

Further simplification of the above expression for the pre-conditionedpilots may be achieved based on the properties of the elements of theunitary Jones matrix. This step is not required, but may be used toachieve further processing simplification in cases where the opticalfiber polarization dependent loss (PDL) and Polarization mode dispersion(PMD) are small, for example.

Since  S_(X)^(H) ∼ J₁₁ = u; S_(X)^(V) ∼ J₁₂ = v;S_(Y)^(H) ∼ J₂₁ = −v^(*); S_(Y)^(V) ∼ J₂₂ = u^(*), thenS_(X)^(H) = S_(Y)^(V^(*)) and S_(X)^(V) = −S_(Y)^(H^(*)),

and following the symmetry among the elements of the S matrix:

$\begin{matrix}{{S_{11} = S_{44}},S_{22}} \\{{= S_{33}},S_{23}} \\{= {S_{32,}^{*}S_{14}}} \\{= {S_{41,}^{*}S_{12}}} \\{= S_{21}^{*}} \\{= {- S_{34}}} \\{= {{- S_{43,}^{*}}S_{13}}} \\{= S_{31}^{*}} \\{= {- S_{24}}} \\{= {- S_{42}^{*}}}\end{matrix}$

This leaves only two real (for example, S₁₁ and S₂₂) and four complex(for example, S₁₂, S₁₃, S₁₄, and S₂₃) independent constant coefficientsout of the 16 original S matrix elements to be calculated and this maybe used for pre-conditioning of the pilots:

P _(X) ^(H) *>=S ₁₁ P _(X) ^(H) *+S ₁₂ P _(X) ^(V) *+S ₁₃ P _(Y) ^(H)*+S ₁₄ P _(Y) ^(V)*

P _(X) ^(V) *>=S ₁₂ *P _(X) ^(H) *+S ₂₂ P _(X) ^(V) *+S ₂₃ P _(Y) ^(H)*−S ₁₃ P _(Y) ^(V)*

P _(Y) ^(H) *>=S ₁₃ *P _(X) ^(H) *+S ₂₃ *P _(X) ^(V) *+S ₂₂ P _(Y) ^(H)*−S ₁₂ P _(Y) ^(V)*

P _(Y) ^(V) *>=S ₁₄ *P _(X) ^(H) *−S ₁₃ P _(X) ^(V) *−S ₁₂ *P _(Y) ^(V)*−S ₁₁ *P _(Y) ^(V)*  (11)

The pre-conditioned pilots <P_(X,Y) ^(H,V)*> obtained from the aboveEquations (7), (10), and (11) have their static parts averaged/filtered,while the common dynamic parts are still intact. This is why thepre-conditioned pilot traces form non-collapsing annular rings withsimilar aspect ratios in the complex plane 110, which are illustrated inFIG. 6, along with the raw pilot traces 100. Pre-conditioning isperformed by averaging over 10 microseconds or, equivalently, byfiltering within 100 kHz, for example, accommodating for the fastest ofthe polarization evolution dynamics.

The pre-conditioning of the pilots guarantees that their annular tracesdo not collapse and, hence, renders the application of the amplitudestabilization of the common dynamic phasor amplitude in accordance withEquation (3) safe and justified:

$\begin{matrix}{{{\operatorname{<<}P_{X,Y}^{H,V^{*}}}\operatorname{>>}{= \; < {{< P_{X,Y}^{H,V^{*}} >}} > {\times \frac{< P_{X,Y}^{H,V^{*}} >}{{< P_{X,Y}^{H,V^{*}} >}}}}},} & (12)\end{matrix}$

The result of this final step of the pilot processing producesamplitude-stabilized circular pilot phasors <P_(X,Y) ^(H,V)*> 110, asillustrated.

The pilot traces of FIG. 6 are normalized to eliminate irrelevantconstants and reduced down to their corresponding Jones matrix elementsfor clarity. This allows for the illustration of the accuracy of theprocedures of the present disclosure by comparing the a priori knownJones matrix elements (the top line of the plot titles and + on theplots) with ones extracted from the raw pilots (the bottom line of theplot titles and x on the plots). The difference is very small (i.e. thestandard deviation between them is 0.001), resulting in perfectposteriori extraction of the Jones matrix elements—crucial for correctpolarization de-multiplexing and elimination of polarization crosstalk.

The ultimate test of the noise tolerance of the proposed pilotaveraging/filtering technique is the reduction of the ASE inflicted Qpenalty with respect to the case of the raw pilots. Such cases weresimulated for the pilot assisted optical PDM OFDM offset QPSK signalsillustrated in FIG. 2. Baud rate was 4 Gb/s and the pilot LPF had 400MHz bandwidth. Simulated Q in dB is plotted in FIG. 7 when computedusing differently conditioned pilots: corresponding to the raw pilots(Q=12 dB) 50, pre-conditioned static parts of the pilots (Q=13.7 dB) 52,additionally stabilized phasor amplitudes of the pre-conditioned pilots(Q=14.1 dB) 54, and artificially noiseless pilots (no ASE noise addedunder the pilots, purely for benchmarking purposes) 56.

These simulations were performed with signal-to-LO laser frequencyoffset randomly chosen between −50 MHz and +50 MHz, both lasers with 1MHz linewidth, and random fiber Jones matrix elements.

It is evident from FIG. 7 that pre-conditioning the static parts of thepilots increases Q by 1.7 dB. Additional stabilization of the pilotphasor amplitudes raises the Q increment to 2.1 dB, just less than 0.4dB short of the ultimate limit obtainable with noiseless pilots. Evenwith a moderate pilot averaging time of 0.25 microsecond (equivalent tofiltering within 4 MHz) the standard deviation of the estimated Jonesmatrix elements from their a priori known values from simulations isonly 0.006, resulting in good polarization de-multiplexing of thereceived signals and negligible polarization crosstalk. Almost all ofthe pilot related ASE Q penalty was recovered as a result of applicationof the pilot averaging/filtering procedures. Small fractions of the dBshortage of the ultimate limit corresponding to the noiseless pilots maybe explained by ASE contributing the azymuthal component of the pilot,thus generating additional phase noise in the phasor indistinguishablefrom the laser phase noise and hence practically unfixable.

Thus, the present disclosure resolves a critical contradiction in pilotbased signal processing: the need for narrowband filtering to mitigateASE noise penalties, and the need for significant spectral bandwidth tomitigate laser frequency, laser linewidth, and nonlinear impairments.The uniqueness of the disclosed techniques is in the non-trivialimplementation of pilot averaging/filtering techniques, allowing for arigorous reduction in noise via averaging/filtering, while maintainingbroadband phase noise cancelling capabilities. The distinctiveaveraging/filtering of the static pilot components andaveraging/filtering of the dynamic phasor magnitude are accomplishedwithout the need for the overburdening labor of separating them beforeand re-combining them after the averaging/filtering procedures.

The present disclosure is directly applicable to the coherent signalprocessing of optical signals, regardless of the specific implementationor modulation format, and finds specific applicability with modulationformats with denser constellations, i.e. ones requiring the eliminationof phase noise, as well as nonlinear impairments. Transport andswitching platforms benefit from improved performance, and substantialhardware density increases afforded by much lower signal processingpower consumption. The same approach is applicable in the wirelessindustry, which uses OFDM-type communication links

Although the present disclosure has been illustrated and described withreference to preferred embodiments and specific examples thereof, itwill be readily apparent to those of ordinary skill in the art thatother embodiments and examples may perform like functions and/or achievelike results. All such equivalent embodiments and examples are withinthe spirit and scope of the present disclosure, are contemplatedthereby, and are intended to be covered by the following claims.

What is claimed is:
 1. A method for noise tolerant signal processing ina pilot assisted data receiver, comprising: given received pilots withcommon pilot components and individual pilot components, computingcoefficients associated with the individual pilot components of thereceived pilots; and applying the computed coefficients to the receivedpilots to obtain conditioned pilots.
 2. The method of claim 1, whereinthe individual pilot components result from relatively slow changes ofthe received pilots relative to the common pilot components.
 3. Themethod of claim 1, wherein the common pilot components result fromrelatively fast changes of the received pilots relative to theindividual pilot components.
 4. The method of claim 1, furthercomprising removing noise from the individual pilot components byaveraging/filtering the individual pilot components.
 5. The method ofclaim 4, wherein the averaging/filtering the individual pilot componentscomprises one of averaging the individual pilot components in a timedomain and low pass filtering the individual components in a frequencydomain.
 6. The method of claim 1, further comprising preserving thecommon pilot components using wideband signal processing.
 7. The methodof claim 1, further comprising down converting the received pilots to abaseband.
 8. The method of claim 1, wherein applying the computedcoefficients to the received pilots to obtain the conditioned pilotscomprises deriving a linear combination of the pilot components withweights corresponding to the computed coefficients.
 9. The method ofclaim 1, further comprising converting the conditioned pilots to aconstant amplitude phasor by removing associated noise related toamplitude fluctuations.
 10. A pilot assisted data receiver for noisetolerant signal processing, comprising: a processing block operable for,given received pilots with common pilot components and individual pilotcomponents, computing coefficients associated with the individual pilotcomponents of the received pilots; and a processing block operable forapplying the computed coefficients to the received pilots to obtainconditioned pilots.
 11. The data receiver of claim 10, wherein theindividual pilot components result from relatively slow changes of thereceived pilots relative to the common pilot components.
 12. The datareceiver of claim 10, wherein the common pilot components result fromrelatively fast changes of the received pilots relative to theindividual pilot components.
 13. The data receiver of claim 10, furthercomprising a processing block operable for removing noise from theindividual pilot components by averaging/filtering the individual pilotcomponents.
 14. The data receiver of claim 13, wherein theaveraging/filtering the individual pilot components comprises one ofaveraging the individual pilot components in a time domain and low passfiltering the individual components in a frequency domain.
 15. The datareceiver of claim 10, further comprising a processing block operable forpreserving the common pilot components using wideband signal processing.16. The data receiver of claim 10, further comprising a processing blockoperable for down converting the received pilots to a baseband.
 17. Thedata receiver of claim 10, wherein applying the computed coefficients tothe received pilots to obtain the conditioned pilots comprises derivinga linear combination of the pilot components with weights correspondingto the computed coefficients.
 18. The data receiver of claim 10, furthercomprising a processing block operable for converting the conditionedpilots to a constant amplitude phasor by removing associated noiserelated to amplitude fluctuations.
 19. A method for noise tolerantsignal processing in a pilot assisted data receiver, comprising: downconverting received pilots to a baseband; given received pilots withcommon pilot components and individual pilot components, computingcoefficients associated with the individual pilot components of thereceived pilots; removing noise from the individual pilot components byaveraging/filtering the individual pilot components; preserving thecommon pilot components using wideband signal processing; applying thecomputed coefficients to the received pilots to obtain conditionedpilots; and converting the conditioned pilots to a constant amplitudephasor by removing associated noise related to amplitude fluctuations.20. The method of claim 19, wherein the averaging/filtering theindividual pilot components comprises one of averaging the individualpilot components in a time domain and low pass filtering the individualcomponents in a frequency domain, and wherein applying the computedcoefficients to the received pilots to obtain the conditioned pilotscomprises deriving a linear combination of the pilot components withweights corresponding to the computed coefficients.